university of antwerp us to visualize the ﬁt of a whole collection of multivariate copulas at...

DEPARTMENT OF ACCOUNTING & FINANCE

A new graphical tool for copula selection

Frederik Michiels & Ann De Schepper

UNIVERSITY OF ANTWERP Faculty of Applied Economics

Stadscampus Prinsstraat 13, B.213 BE-2000 Antwerpen Tel. +32 (0)3 265 40 32 Fax +32 (0)3 265 47 99 http://www.ua.ac.be/tew

FACULTY OF APPLIED ECONOMICS

DEPARTMENT OF ACCOUNTING & FINANCE


Frederik Michiels & Ann De Schepper

RESEARCH PAPER 2010-004 MAY 2010

University of Antwerp, City Campus, Prinsstraat 13, B-2000 Antwerp, Belgium Research Administration – room B.213

phone: (32) 3 265 40 32 fax: (32) 3 265 47 99

e-mail: [email protected]

The papers can be also found at our website: www.ua.ac.be/tew (research > working papers) &

www.repec.org/ (Research papers in economics - REPEC)

D/2010/1169/004

mailto:[email protected]�

http://www.ua.ac.be/tew�

http://www.repec.org/�


Frederik Michiels∗

Faculty of Applied Economics, University of Antwerp, Prinsstraat 13, Antwerp, Belgium

Ann De Schepper

Faculty of Applied Economics & StatUa Statistics Center, University of Antwerp, Prinsstraat 13,

Antwerp, Belgium

Abstract

The selection of copulas is an important aspect of dependence modeling. In many practical

applications, only a limited number of copulas is tested, and the modeling applications usually

are restricted to the bivariate case. One explanation is the fact that no graphical copula tool

exist which allows to assess the goodness-of-fit of a large set of (possible higher dimensional)

copula functions at once. This paper pursues to overcome this problem by developing a

new graphical tool for the copula selection, based on a statistical analysis technique called

‘principal coordinate analysis’. The advantage is threefold. In the first place, when projecting

the empirical copula of a modeling application on a two-dimensional copula space, it allows

us to visualize the fit of a whole collection of multivariate copulas at once. Secondly, the

visual tool allows to identify ‘search’ directions for potential fit improvements (e.g. through

the use of copula transforms). Finally, in the bivariate case the tool makes it also possible

to give a two-dimensional visual overview of a large number of known copula families, for a

common concordance value, leading to a better understanding and a more efficient use of the

different copula families. The practical use of the new graphical tool is illustrated for two

two-dimensional and two three-dimensional fitting applications.

Keywords: copulas, copula selection, Kendall’s tau, graphical tool, multidimensional scaling,

dependence models

∗[email protected].

A new graphical tool for copula selection 2

1 Introduction

In a recent bibliometric study by Genest et al. (2009), one of the conclusions of the authors is that

there is a clear need for the further exploration of graphical tools for the selection and validation

of copula models. A drawback of the copula visualization tools that are used most commonly, e.g.

the level-curve representation or a density-cure plot, is that they become inappropriate when it

comes to visualize the goodness-of-fit of a collection of copulas. The alternative of measuring and

visualizing the fit of every copula separately is rather tedious and unpractical.

A copula visualization tool which gives a clear overview of the proximity of a (large) collection

of copulas in a dependence modeling application is, to the best of our knowledge, non-existent.

Nevertheless, from a practical point of view this is an absolute necessity, as it facilitates the

consideration of a large number of possible copulas and therefore make the copula selection less

arbitrary.

In order to create a two-dimensional representation of a copula space, we use a technique called

principal coordinate analysis. We start by creating an Euclidean distance matrix, based on the

dissimilarities between the values of the cumulative distribution function (cdf) for different copulas,

evaluated over a fine grid. Next, we construct a configuration of points in two dimensions, where

the Euclidean distances between the different copulas correspond to the original distance matrix.

In order to check whether the 2D representation is appropriate, the goodness-of-fit of the two

dimensions can be calculated using the stress-function.

The copula selection tool we develop can be used to visualize both bivariate and multivariate

goodness-of-fit problems. For the bivariate case we present an overview of 21 well-known one-

parameter copula families. The overview is presented as a representation of ‘comparable’ copulas

for a common Kendall’s τ value, for 10 % intervals, and can be used as a copula guide for selection

or study purposes. The use of the selection tool is shown by means of two examples of well-known

data sets. For the multivariate case the selection tool is used to represent the fit of a collection

of 18 multivariate copula families in two three-dimensional fitting examples. Finally, the copula

selection tool has the extra advantage over existing graphical tools in that it allows to visualize

search directions for potential fit improvements.

The paper is organized as follows. In Section 2 an overview of existing (bivariate) graphical

tools is presented and the new copula selection tool is developed. In Section 3 the bivariate case is

discussed, including the copula guide and two fitting examples. Section 4 discusses the multivariate

case and finally, Section 5 contains a conclusion.


2 A new graphical tool for copulas

2.1 Graphical tools: an overview

Recall that a copula C is a multivariate cumulative distribution function (cdf) on the unit n-cube

[0, 1]n with uniform margins. By virtue of Sklar (1959), copulas provide a flexible tool for the

construction of a n-dimensional cdf H from continuous margins F as H(x) = C(F (x)). In the

continuous case, such a copula is unique. Consequently, multivariate goodness-of-fit problems can

be reduced to copula selection problems.

To our knowledge, the availability of graphical tools which can be used for the problem of

(bivariate) copula selection is rather limited. We now provide a short overview.

1. Bivariate copulas can well be represented by displaying a 3D image of their cdf or probability

density function (pdf), or plotting a 2D level curve representation. Although this technique works

fine for the investigation of a single copula, it becomes unpractical for the comparison of the fit

of a whole collection of copulas and it clearly is restricted to the bivariate case. In Figure 1 the

four representation forms are depicted. Note that for the cdf level curve representation (Figure

1, upper right panel), a possible strategy would be to compare the level curves to those of the

co-monotonic, independent and counter-monotonic copulas.

u1

u2

0 0.5 10

0.5

1

u1

u2

0 0.5 10

0.5

1

00.5

1

00.5

10

0.5

1

u1u2

00.5

1

00.5

10

1

2

u1u2

Figure 1: Representation of a Gaussian copula, ρ = 50%, 3D cdf (upper right), 2D cdf level curve

(upper left), 3D pdf contours (lower right), 2D pdf level curve (lower left).

2. For bivariate Archimedean copulas, Genest and Rivest (1993) show that the goodness-of-fit

can be assessed using quantile-quantile-plots based on the bivariate probability integral transform

K(t) = P (C(U1, U2) ≤ t), for any t ∈ [0, 1], and they develop a procedure to obtain K empirically.

In a recent paper we also stress the visual advantage of K in recognizing key dependence character-


istics of an Archimedean copula represented in the lambda function λ(t) = t − K(t), see Michiels

et al. (2010). In Figure 2 an example is provided; note that λ(t) = t ln(t) corresponds to the

independent copula. The downside, however, is that K and λ are only parametrically available for

Archimedean copulas, and just as it holds for contour diagrams, the comparison of dissimilarities

among several copula families can be unclear and subjective.

0 0.5 10

0.5

1

u1

u 2

(U1,U

2) ∼ Cλ

0 0.5 1−1

−0.5

0

t

λ(t)

The function λλ(t)=tln(t)

Figure 2: Illustration of the lambda function for a copula with λ(t) = t(t−1)(7.71t2−10.31t+3.5),

τ = 58.13%, λL = 8.84%, λU = 13.39%.

3. A final graphical tool for picturing bivariate copulas is based on conditional expectation curves,

see e.g. Nikoloulopoulos and Karlis (2008). The visualizations can provide more insight into

similarities between copula families. In Figure 3 an example is given. For higher dimensional

copulas, the possible regression curves augment quickly (certainly for non-exchangeable copulas),

and clearly one graph would not suffice to get a clear image of the copula. Furthermore, displaying

a large collection of copulas in this way, again, becomes confusing and unpractical.

0 0.5 10

0.5

1Clayton

u2

E(U

1|U2=

u 2)

0 0.5 10

0.5

1Gumbel−Hougaard

u2

E(U

1|U2=

u 2)

Figure 3: Conditional expectation curves, the Clayton copula vs. the Gumbel-Hougaard copula,

τ = 20%.

In the following section, we show how a visualization based on principal coordinate analysis

is much more convenient in order to compare the appropriateness of a large collection of copulas,

displayed together, and how it can be used straightforwardly for goodness-of-fit purposes. This

visual tool can easily be extended to the multivariate case, see section 4.


2.2 Description of our new graphical tool

The graphical tool developed in this contribution heavily relies on principal coordinate analysis.

Principal coordinate analysis or metric multidimensional scaling is a multivariate statistical data

analysis technique used to gain insight in the underlying relations which are present in a multivari-

ate dataset. In the context of our investigation, the multivariate dataset corresponds to a collection

of copulas with cdf-values. The main objective of the technique is to provide a low-dimensional

representation of high-dimensional data, in such a way that the distortion caused by a reduction

in dimensionality is minimized.

In this subsection we only briefly recall the main aspects of this technique. We refer to the

seminal papers of Kruskal (1964a,b) for a concise description, or to Kruskal and Wish (1978) or

Borg and Groenen (1997) for a more elaborate analysis. In the sequel, we adopt the same notations

as Johnson and Wichern (1992).

Multidimensional scaling techniques deal with the following problem: For a set of observed

similarities between every pair of n copulas, find a representation of the items in few dimensions

such that the inter-item proximities nearly match the original similarities. In this analysis we use

classical multidimensional scaling, which means that we use the Euclidean distance as a proximity

measure in the final low-dimensional configuration.

We now come to the basic algorithm. Departing from a set of n copulas C1, C2, · · · , Cn

(in cdf form) we compute the (simulated) dissimilarity between the ith and jth copula through

d(i,j) =√

(Ci − Cj)′(Ci − Cj), the Euclidean distance. For the examples, we use 1000 random

copula observations, as this yields stable results (i.e. simulated distance corresponds to theoretical

distance). The Euclidean distance is chosen in order to give every cdf-observation equal weight.

For the n copulas we obtain M = n(n − 1)/2 distances between pairs of different copulas.

Assuming no ties, these distances can be arranged in a strictly ascending order as

d(i1,j1) < d(i2,j2) < · · · < d(iM ,jM ) (1)

with 1 ≤ ik 6= jk ≤ n (k = 1, · · · ,M) where d(iM ,jM ) is the largest of the M dissimilarities and the

subscript (i1, j1) indicates the pair of copulas which is most similar. The challenge now is to find

a q-dimensional configuration of the n items (q < n) such that in terms of the q new dimensions,

the distances d(q)(i,j) between pairs of copulas match the ordering in (1). If the new distances are

laid out in a manner corresponding to that ordering, a perfect match occurs when

d(q)(i1,j1)

< d(q)(i2,j2)

< · · · < d(q)(iM ,jM ). (2)


For our copula applications we strictly want to use 2D representations and consequently we fix

q = 2. The coordinates (x, y) attached to the new distances d(2) are then found by minimizing the

so called ‘stress’ function over the M coordinates (xi1 , yi1), · · · , (xiM, yiM

) which is defined as

Stress =

∑Mk=1 (d

(2)(ik,jk) − d(ik,jk))

2

∑Mk=1 d

(2)(ik,jk)

2

1/2

(3)

with distance d(2)(ik,jk) =

√

(xik− xjk

)2 + (yik− yjk

)2, where ik, jk (k = 1, · · · ,M) are determined

by equation (1).

A stress value of 2.5% is said to provide an excellent fit, see e.g. Kruskal (1964a). Our objective

is to find a two-dimensional representation of the copula test spaces, both for creating a copula

guide and for goodness-of-fit purposes. For the former (see Section 3.2) it can be seen that a nearly

perfect match occurs for all test spaces for 10 % τ intervals and as such, it follows that optimizing

equation (3) also leads towards the correct ordering of all d(2)’s. However, when comparing the

goodness-of-fit of each member of a whole collection of copulas used as a model for an empirical

copula (see Sections 3.3 and 4.2), optimizing equation (3) may lead towards violations against

equation (2), which becomes problematic. Hence, we propose to add a restriction on the scaling

technique for goodness-of-fit applications, with respect to the distances between a copula from the

test space and the empirical ocula. Let

d(1,emp) < d(2,emp) < · · · < d(n,emp) (4)

be the ordering of the distances between the n parametric copulas from the test space and the

empirical copula. Then, in order to assure that equation (4) is always satisfied during the mini-

mization process, add the following restriction to (3):

d(2)(i,emp) < d

(2)(j,emp) if d(i,emp) < d(j,emp) (5)

where d(2)(i,emp) =

√

(xi − xemp)2 + (yi − yemp)2 .

As mentioned earlier, the representation in this contribution is based on the Euclidean distance.

Consequently, the goodness-of-fit metric will be Euclidean too. This assumption, however, can be

relaxed, one can take other distance measures like e.g. the Kolmorov-Smirnoff distance or the

Cramer-von Mises metric. Without entering into details, the latter will yield more or less the

same results as it merely differs in taking the average of all distinct distance pairs. The Kolmorov-

Smirnoff will yield different results, but in our opinion it is too sensitive for outliers and therefore

not the best choice. Finally, note than when necessary a more robust distance metric than the

Euclidean distance can be applied.


2.3 Utilisation

The main idea of our graphical tool consists of making a 2-dimensional representation of the

empirical copula together with a test space, which is a set of workable theoretical copula functions.

In the bivariate case, the most obvious approach is to choose for this set of copula models the

whole comparable test space, corresponding to the estimated τ -value of the empirical copula (see

Michiels and De Schepper (2008)). In the multivariate case, we can use a maximum likelihood

approach in order to estimate the parameters of some copula models based on the empirical data,

and we then define the set of copula functions as the set of these comparable copula models.

In the ideal situation, the stress value is smaller than 2.5% for q = 2, allowing to work with

a 2D-representation of the copula test space. Then plot the empirical copula in the same plane

together with the entire test space; in order to model the dependence in the data, one can choose

for the copula with the highest degree of similarity with the empirical copula. This approach will

be used for the examples in section 3.3 and 4.2.

Note that the 2D-representation of comparable copula functions also admits mutual compar-

isons of comparable copula functions. This will be illustrated for the bivariate case in section 3.2.

We refer to appendix A for more information on the investigated copulas and on Archimedean

copulas in particular.

Finally, the visual tool permits the identification of search directions for potential goodness-of-

fit improvements and the inclusion of these improvements on the 2D-representation (see sections

3.3 and 4.2). These improvements can be achieved by making use of transforms preserving or

changing certain aspects of the copula model in such a way that a better fit is obtained. We

illustrate this possibility with two examples in the bivariate case.

A first strategy for improving the goodness-of-fit could be to create a non-exchangeable alter-

native of the best fit copula from our set. A straightforward technique can be found in Genest

et al. (1998), where the following two-parameter transform is proposed:

Cκ,η(u, v) = u1−κv1−ηC(uκ, vη) (6)

where 0 < κ, η < 1.

Another strategy consists of looking for goodness-of-fit improvements by holding some aspects

of the best fit copula constant. In this respect, we first summarize a result, introduced in Michiels

and De Schepper (2009), which allows to construct τ -preserving transforms, starting from an

arbitrary Archimedean copula function. Let fθ : [0, 1] 7→ [0, 1] be an increasing bijection with


parameter vector θ. A τ -preserving transform is given by

λt(t) =dfθ

dt(t) · λ(fθ(t)) (7)

where the range for θ is determined using feasibility conditions on λ, which are λ(0) = 0, λ(1) = 0

and λ(t) < 0,dλ

dt(t) < 1 for all t ∈ (0, 1).

This means that, for any Archimedean copula and for any increasing bijection on [0, 1], we

can create a new copula, with the same value for the concordance parameter, but which differs

from the original copula in a certain way. When the transformed copula functions are depicted

on the 2D-representation as well, together with the empirical copula and together with the other

copula functions of the comparable test space, it allows the choice of an appropriate transform

and possibly an appropriate value for the potential parameter of the transforming function fθ,

corresponding to a better overall fit. This technique will also be illustrated in section 3.3.

As an extra benefit of this approach, note that the transform can be chosen such that next

to the concordance value, also the tail dependence coefficients of the original copula model are

preserved. This will be the case when the function fθ : [0, 1] 7→ [0, 1] satisfies the extra conditions

f ′

θ(0) = 1, f ′

θ(1) = 1, with f ′′

θ (0) bounded.

3 The Bivariate Case

As stated in the introduction, the contribution in the bivariate case is threefold. Using our graphical

tool we present a copula guide of 21 well-known one-parameter copula families with potential

practical use. Secondly, we illustrate the benefits of our tool for copula selection for two fitting

applications, and finally we search for fit improvements.

3.1 Selection of copulas

In the remainder of this section the following copula families are used (their functional forms are

presented in the first appendix):

• Archimedean class: 14 strict Archimedean families, including the popular Clayton, Frank,

Gumbel-Hougaard, Joe and Ali-Mikhail-Haq families (C1, · · · , C14);

• Extreme value families: Galambos and Husler-Reiss family or Gumbel-Hougaard family

(C15, C16);

• Meta-elliptical class: Normal, Student’s t family (C17, C18);


• Other families: Farlie-Gumbel-Morgenstern (FGM) family, Plackett family and Raftery fam-

ily (C19, C20, C21).

For more detailed information about the Archimedean copula class, please consult e.g. Chapter

4 in Nelsen (2006) and McNeil and Neslehova (2009); for the meta-elliptical class, see Fang and

Fang (2002) and for extreme value class, we refer to Joe (1997).

In the bivariate case the copula parameters can well be interpreted in terms of concordance.

Indeed, for Kendall’s tau we have τ = 4∫ 1

0

∫ 1

0C(u1, u2)dC(u1, u2) − 1 and for the coefficients of tail

dependence λL = limu→11−2u+C(u,u)

1−u and λU = limu→0C(u,u)

u . Note that for the Archimedean

class, these quantities can be written straightforwardly by means of the λ-function: τ = 1 +

4∫ 1

0λ(t)dt, λL = 2λ′(0+) and λU = 2 − 2λ′(1−), see Michiels et al. (2010).

A bivariate copula C is called exchangeable if C(u1, u2) = C(u2, u1), it is reflection symmetric

if C(u1, u2) = CS = u1 + u2 − 1 + C(1 − u1, 1 − u2), where CS is called the survival copula, and

it exhibits the extreme value property if C(uk1 , uk

2) =(

C(u1, u2))k

for all k > 0.

Remark that all 21 copula families considered here are in fact exchangeable. In order to

get an idea of the diversity of this collection of copula families we present an overview of some

properties of the 21 copula families in Table 1. Columns two and three contain the coefficients of

tail dependence in function of the copula parameter θ. Note that the Student’s t copula has two

parameters, the correlation coefficient ρ and the degrees of freedom v. The fourth and fifth column

display whether or not a copula family possesses the reflection symmetry and/or the extreme value

property. Finally, the last column indicates the copula dependence range in terms of Kendall’s τ .

3.2 Copula guide

We now visualize the relative differences between one-parameter copula families for a common

concordance value (Kendall’s tau), using multidimensional scaling. The technique is applied to the

21 copula families introduced earlier, each copula being represented as a vector containing 1000

cdf data points. The goal is to map each of the comparable test spaces (i.e. a set of copulas

adequate to model a certain degree of dependence, see Michiels and De Schepper (2008)) into the

two-dimensional space, by assigning a point in the (x, y)-plane to each copula of the test space.

For each analysis we check the stress function to see whether a 2D representation is feasible. The

stress values are summarized in Table 2. It is clear that these stress values indicate an excellent

goodness-of-fit for all test spaces.


family λL λU Reflection Extreme Kendall’s τ

symmetry value range

1 2−1/θ 0 [0, 1]

2 0 0 [ 12(5 − 8 ln 2, 1

3]

3 0 2 − 21/θ √[0, 1]

4 0 0√

[−1, 1]

5 0 2 − 21/θ [0, 1]

6 0 0 [−0.36, 0]

7 0 0 [−0.18, 0]

8 2−1/θ 2 − 21/θ [ 13, 1]

9 0 0 [−0.36, 1]

10 12

2 − 21/θ [ 13, 1]

11 12

0 [−1, 13]

12 0 0 [−0.61, 1]

13 1 0 [ 13, 1]

14 1 0 [0, 1]

15 0 2−1/θ √[0, 1]

16 0 2 − 2Φ(1/θ)√

[0, 1]

17 0 0√

[−1, 1]

18 2tv+1

(

−

√(v+1)

√(1−ρ)√

(1+ρ)

) √[−1, 1]

19 0 0√

[−29

, 29]

20 0 0√

[−1, 1]

21 2θ1+θ

0 [0, 1]

Table 1: Characteristics for 21 copula families.

τ stress value

-90% < 10−3

-80% < 10−3

-70% < 10−3

-60% < 10−3

-50% < 10−3

-40% < 10−3

-30% 0.12%

-20% < 10−3

-10% 0.22%

τ stress value

10% 0.55%

20% 1.01%

30% 0.91%

40% 0.85%

50% 0.97%

60% 0.98%

70% 0.97%

80% 1.59%

90% 1.83%

Table 2: Stress values for principal coordinate analyses for different values of τ .


Negative dependence

In Figure 8, a two-dimensional mapping of the copula families accounting for negative dependence

is shown, for nine cases corresponding to τ ∈ {−90%, −80%, −70%, −60%, −50%, −40%, −30%,

−20%, −10%}. The following observations can be made:

• For τ ∈ {−90%,−80%,−70%}, the feasible copula families are the Archimedean copulas

{C4, C11}, the meta-elliptical copulas {C17, C18} and copula C20. Note that C18 is represented

for 3 degrees of freedom. As all bivariate copulas share the same lower bound, it is obvious

that the relative differences augment for increasing values of τ , see panel (a)-(b)-(c).

• For τ ∈ {−60%,−50%,−40%}, the set of comparable copulas is extended with the Archimedean

copula C12. In general, two main groups remain, but they get closer. See panel (d)-(e)-(f).

One clear group can be identified, consisting of the copulas with reflection symmetry. On

the other hand, copulas C11 and C12 are clear outliers. The asymmetric behavior of C11 is

clear, due to the fact that it has constant lower tail dependence, the asymmetric behavior of

C12 can be deduced from the plots and is more or less opposite to that of C11, although the

former has no upper tail dependence(1).

• For τ ∈ {−30%,−20%,−10%}, the comparable test space is enlarged again, now with copulas

C6, C9, C19, C7 and C2 (in that order). See panel (g)-(h)-(i).

Remark that as τ approaches zero, the relative differences between the copula families en-

tailing independence become smaller and smaller. The clear outliers are C11 and C18, and

this can be explained by the fact that these copula families do not include the independence

copula.

Positive dependence

In Figure 9, a two-dimensional mapping of the copula families accounting for positive dependence

is shown, for nine cases corresponding to τ ∈ {10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%}.

The following observations can be made:

• For τ ∈ {10%, 20%, 30%}, the comparable test space is very large, with the Archimedean

copulas {C1, C2, C3, C4, C5, C6, C9, C11, C12, C14}, the extreme value copulas {C15, C16}, the

meta-elliptical copulas {C17, C18} and the other copula families {C19, C20, C21}, with copula

(1)This fact can also be assured by making λ-plots.


C19 disappearing for τ > 20%, see panel (a)-(b)-(c). We also added the survival versions of

C1 and C3, for reasons of comparison.

We observe that for τ > 20% three regions can be identified: on the left we have copulas

with lower tail dependence, in the centre copulas with symmetric tail dependence, on the

right copulas with only upper tail dependence. Remark that with increasing τ -values, the

group {C1, C2, C11, C14} can be recognized, all sharing the element C(u1, u2) = u1u2

u1+u2−u1u2

for τ = 13 .

Note also the high similarity between the extreme value copulas {C3, C15, C16}, which is

apparent throughout the whole analysis: they share the extreme value property, they are also

exchangeable and their upper tail dependence parameter can be approximated as λU ≈ 21−τ .

• For τ ∈ {40%, 50%, 60%}, a new group of Archimedean copulas enters the test space, namely

{C8, C10, C13}, see panel (d)-(e)-(f).

Notice the strong similarity between copulas C13 and C14 (due to their exceptional lower tail

dependence property), C1 and C21 (both varying lower tail dependence), C10 and C18 (both

upper and lower tail dependence), and CS1 and C5 (relative high upper tail dependence).

• For τ ∈ {70%, 80%, 90%}, relative distances become smaller as all copulas converge to the

Frechet upper bound; therefore, for panel (i), the scale is different.

As τ approaches 1, the most important and only diversifying factor is λL.

How can this copula guide be used in a practical fitting problem?

Suppose that for a particular modeling problem, an estimate for the overall concordance param-

eter τ has been calculated, and suppose that one wants to choose one or more copulas to model the

dependence structure in the data. A good approach would be to start from the pictures in Figures

8-9, in particular the representation corresponding to the situation with the closest τ -value, and

to choose some copula families with large dissimilarities. After testing the goodness-of-fit for each

of these three or four copula families, one can refine the search procedure, by exploring some more

copula families with a higher degree of similarity with the best copula family from the first loop.

3.3 Empirical illustrations

In the previous section we showed that principal coordinate analysis allows to represent comparable

test spaces in a two-dimensional space, and their relative differences can be interpreted using known

copula properties. We now proceed with a practical application of the visualization tool. The 2D


−0.8 −0.4 0 0.4 0.8−0.8

−0.4

0

0.4

0.8

C1

C1SC

2

C3

C3S

C4

C5

C9

C11

C12

C14

C20

C21 C

17

C18

C15C

16

Cemp

−0.2 −0.1 0 0.1 0.2−0.2

−0.1

0

0.1

0.2

C3

C15

C16

Cemp

C3A

Figure 4: 2D representation goodness-of-fit Loss-ALAE dependence structure (left), 2D represen-

tation of goodness-of-fit improvement (right).

mapping allows for the representation of the goodness-of-fit of a complete comparable test space

(i.e. a collection of copulas). Indeed, plotting the empirical copula Cemp(2) into a comparable

test space, enables the investigation of the relative distances between the empirical copula and all

copulas of the test space at once. Afterwards, we can look for a search strategy to move closer to

Cemp.

3.3.1 The Loss-ALEA data set of Frees and Valdez (1998)

As a first example we take the well-known Loss-ALEA data set studied by Frees and Valdez (1998).

The data comprise 1,500 general liability claims (expressed in USD) randomly chosen from late

settlement lags, and were provided by Insurance Services Office, Inc. Each claim consists of an

indemnity payment (loss) and an allocated loss adjustment expense (ALAE). Here ALAE are types

of insurance company expenses that are specifically attributable to the settlement of individual

claims such as lawyers’ fees and claims investigation expenses. In order to price an excess-of-loss

reinsurance treaty when the reinsurer shares the claim settlement costs, the dependence between

losses and ALAEs has to be taken into account. Departing from the well-known Clayton, Frank

and Gumbel-Hougaard test space, Frees and Valdez (1998) conclude that the Gumbel-Hougaard

family provides the best fit.

The observed overall dependence is estimated using a rank based estimator of Kendall’s τ

yielding τ = 31.54%. If we apply our approach to this example, by visualizing the empirical copula

in the two-dimensional projection of the comparable test space associated with τ , we obtain Figure

4 (left panel). This graph displays the performance of a large set of copulas, and it can be seen

(2)For a definition of the empirical copula, see appendix C.


−0.5 −0.25 0 0.25 0.5−0.5

−0.25

0

0.25

0.5

C1 C

1S

C3

C3S

C4

C5

C8

C9

C10

C12

C13C

14

C20C

21C

17C

18

Cemp

C15

C16

−0.5 −0.25 0 0.25 0.5−0.5

−0.25

0

0.25

0.5

C1

C1SC

3C

3S

C4

C5

C8

C9

C10

C12

C13

C14

C20

C21

C17

C18

Cemp

C15

C16

C4A

C4T

λL

λL

λU

λU

τ and λU, λ

L

preservingdirection

Figure 5: 2D representation goodness-of-fit caesium-scandium dependence structure (left) and

improvement strategy for goodness-of-fit caesium-scandium dependence structure (right).

immediately that the distance between Cemp and the Gumbel-Hougaard family C3 is much smaller

than for the Clayton family C1 or the Frank family C4 – which confirms the choice of Frees and

Valdez. However, we see that next to C3 also the (other) extreme value families C15 and C16

provide a good fit. When taking a closer look (right panel of Figure 4), and adding the best

fit asymmetric version of the Gumbel-Hougaard copula through equation (6) (using a maximum

likelihood approach), it becomes clear that the fit can in fact be slightly improved. From the

symmetric copulas the Galambos copula (C15) provides the best fit, but the asymmetric copula

CA3 performs better.

3.3.2 The uranium-caesium data set from Genest and Rivest (1993)

As a second example we take the data set used in Genest and Rivest (1993) where the authors

present 655 analyses from uranium and caesium log-concentrations (in parts per billion). In their

paper the authors conclude that the Frank copula provides the better fit, compared to the Clayton,

Gumbel-Hougaard family and a two-parameter ‘log copula’ family. We now use our approach to

check whether or not a better fit is possible when using a copula test space of 21 families. The

estimated τ value equals 47.96%. We proceed as in the previous section by visualizing the empirical

copula into the appropriate comparable test space. This is done in Figure 5 (left panel).

The 2D mapping shows that the Frank copula family (C4) indeed provides the best fit amongst

the one-parameter families. However, the best fit distance is quite large (0.2206) and therefore it

can be argued that an improvement of this fit should be investigated. As explained in Section 2.3,

we can look for an improvement of the fit of the Frank copula by choosing an appropriate τ -

preserving transform. In this way, we can search in the neighbourhood of C5 without changing the


overall degree of dependence (and hence, we keep on working in the same comparable test space).

We know from Table 1 that the Frank copula induces no tail dependence. When looking again

at Figure 5 (left panel), we see that the fit for copulas which do have tail dependence is worse. As

a result, we need a transform which is both concordance and tail preserving.

Following the approach as explained in section 2.3, we look for increasing functions f : [0, 1] 7→

[0, 1], with f(0) = 0, f(1) = 1, f ′(0) = 1, f ′(1) = 1 and f ′′(0) bounded.

In order to create a model with sufficient degrees of freedom, we choose for f(·) a polynomial

of order 6. Applying the constraints mentioned above, this leads to the following three-parameter

model:

fθ1,θ2,θ3(t) = t(θ1t

5 + θ2t4 + θ3t

3 − (4θ1 + 3θ2 + 2θ3)t2 + (3θ1 + 2θ2 + θ3)t + 1). (8)

For this choice, there are three shape parameters, denoted by θ1, θ2 and θ3. The parameters are

estimated using the sum of squares principle on the empirical λ-function. The optimal values are

θ1 = −6.9149, θ2 = 19.3122 and θ3 = −18.6477. These values satisfy the feasibility conditions on

the function f . As a result of this transform, the minimal Euclidean distance lowers to 0.1436.

Figure 5 (right panel) shows the transformed copula as CT4 . Note also that the best fit asymmetric

Frank copula (CA4 ) does not yield a significant improvement in comparison with CT

4 .

4 The Multivariate Case

The 2D representation of a set of copula families can straightforwardly be extended to the multi-

variate case, given that the stress function results in acceptable values. We show the use of our

copula selection in two three-dimensional fitting examples.

4.1 Selection of copulas

Unfortunately, it is not possible to extend all 21 bivariate families from section 3.1 from a bivariate

to a multivariate setting. As such, we left out some families because no multivariate extension is

known or possible, or no closed form is available. In order to make up for this loss we added other

valuable multivariate families.

We use the following 18 multivariate copula families, where we kept the same numbering as in

section 3 for the extended copulas:

• Archimedean class: 8 exchangeable Archimedean families, including the popular Clayton,

Frank, Gumbel-Hougaard, Joe and Ali-Mikhail-Haq families (C1, · · · , C5 and C8, · · · , C10)


and 4 Archimedean families with partial symmetry (families M3, ..., M6 as introduced in Joe

(1997));

• Extreme value families: families MM1, MM2 and MM3 defined in Joe (1997);

• Meta-elliptical class: Normal, Student’s t family (C17, C18);

• Other families: multivariate Farlie-Gumbel-Morgenstern (MFGM) family as defined in Nelsen

(2006) (C19).

For more details about these copula families, we refer to the second appendix.

4.2 Empirical illustrations

4.2.1 Financial data set from Michiels and De Schepper (2008)

We use data from Michiels and De Schepper (2008) consisting of 1499 total returns of the S&P

500 Composite Index, the JP Morgan Government Bond Index and the NAREIT All index, over

the period from January 4 2002 until March 13 2008. In Michiels and De Schepper (2008) we

only modeled the bivariate margins. We are now interested in the three-dimensional dependence

structure. In order to get an idea of the dependence present in the data, we first estimate the

concordance matrix, see Table 3.

τ stocks bonds real estate

stocks 1 -0.1721 0.4351

bonds -0.1721 1 -0.0599

real estate 0.4351 -0.0599 1

Table 3: Estimated values of Kendall’s τ for the financial data set.

We show how our graphical tool can be used to get a picture of the goodness-of-fit of the 18

copula families. The parameter estimation is carried out using the maximum likelihood approach.

It appears that for the financial data set two out of three bivariate margins actually contain negative

dependence. In our copula set, the exchangeable (one parameter) Archimedean copula families,

the asymmetric (two parameter) Archimedean families and the (7 parameter) multivariate extreme

value copulas can only model positive dependence, whereas the meta-elliptical copulas and the (4

parameter) MFGM family also allow for pairwise negative dependence.


−0.5 −0.25 0 0.25 0.5−0.5

−0.25

0

0.25

0.5

C1C

4

C3

CM3

CM4

CM5 C

M6

C17

C18

C2

C5

C9

C19

CMM1C

MM2C

MM3

Cemp

Figure 6: 2D representation of three-variate financial data fitting example.

In figure 6 the result of the fitting example can be found. The stress is calibrated on 5.04%

which indicates a good fit. From the graph it follows that the meta-elliptical families C17, C18

outperform all other copulas and that they yield a very good fit. The MFGM model C19 performs

quite good, as indeed in this example the general dependence is rather low and negative pairwise

dependence is present. Next, the multivariate extreme value copulas, CMM1 and CMM3 perform

best. Notice that family CMM2, which has as a special case C1 (see Joe (1997)), actually equals

C1 here. Finally, all symmetric Archimedean copulas (C1, · · · , C5 and C9)(3) perform rather bad;

this is due to the fact that they only have one parameter and can only be used to model positive

dependence, and as a consequence, the estimated parameter values are close to the independence

value.

4.2.2 Extension of the data set from Genest and Rivest (1993)

In the previous example, the data set contained bivariate margins with negative dependence,

which partially explains why all Archimedean copulas (both symmetric and asymmetric) per-

formed rather poorly. We now visualize a dependence modeling example containing only positive

dependence, and we rely once more on the data set from Genest and Rivest (1993). The uranium-

caesium pair actually comes from a multivariate data set introduced in Cook and Johnson (1981)

containing 6 chemical elements. More specifically, we extend the uranium-caesium pair to a three-

dimensional data set including the element potassium. The reason for this choice is the fact

(3)Families C8 and C10 are left out because of their very poor performance.


−0.8 −0.4 0 0.4 0.8−0.8

−0.4

0

0.4

0.8

C1

C2

C3

C4

C5

C9

C17

C18

CM3

CM4

CM5

CM6

C19

CMM1

CMM2

CMM3

Cemp

−0.8 −0.4 0 0.4 0.8−0.8

−0.4

0

0.4

0.8

C1

C2

C3

C4

C5

C9

C17

C18

CM3

CM4

CM5

CM6

C19

CMM1

CMM2

CMM3

Cemp

λU

λL

λL,λ

U

symmetrypresevingdirection

λL,λ

U

asymmetrypresevingdirection

Figure 7: 2D representation of caesium-uranium-potassium fitting example. (left), 2D representa-

tion of goodness-of-fit improvement (right).

that potassium is most correlated with (uranium,caesium) among the remaining elements, with

τ(potassium, ceasium)= 30.40% and τ(potassium,uranium)= 13.45%. The fit is visalized in figure

7, left panel. The stress function yields 3.80%, indicating a very good fit.

This example exhibits a totally different goodness-of-fit picture. Note first of all that the

difference between fitting results for the meta-elliptical copulas and for the other copula families

is not that striking as in the previous example. In fact, an asymmetric version of the Frank

copula CM3 yields the best fit now, although the meta-elliptical copulas come at the second place.

Furthermore, the MFGM copula no longer yields a good fit, as a consequence of its limiting

parameter range. Finally, the copula families entailing (asymmetric) tail dependence do not yield

a good fit, and an improvement of the fit of CM3 should be sought by using tail-preserving and

asymmetry-preserving transforms, see Figure 7, right panel.

5 Conclusion

In this contribution we introduced a new graphical copula tool which serves very well in getting

an idea about the relative similarities and dissimilarities between different copulas. Therefore,

it can also be used as a visual guideline in copula selection problems. The graphical tool is

based on the statistical technique of multidimensional scaling, where a low-dimensional mapping

of a multivariate dataset is provided. We found that a collection of bivariate copulas can be well

represented in a two-dimensional space (a more than excellent fit) and the obtained distances can be

explained using well-known copula properties. We also illustrate the possibilities of this graphical

tool for two particular empirical examples, where we also visualized the optimization scheme based


on the concordance and tail preserving transforms, introduced in Michiels and De Schepper (2009).

Finally, the visual tool can also be used to evaluate the fit of higher dimensional copulas; we

illustrated this by two three-dimensional examples.

Appendix

A. Bivariate copula families

A.1 Archimedean copula families

A strict bivariate Archimedean copula is defined as a function C : [0, 1]2 → [0, 1] : (u1, u2) 7→

C(u1, u2) = ϕ−1(ϕ(u1) + ϕ(u2)), where the generator ϕ : [0, 1] → [0,∞) is continuous, strictly

decreasing and convex with ϕ(0) = +∞ and ϕ(1) = 0.

The λ-function for an Archimedean copula is defined by λ(t) =ϕ(t)

ϕ′(t).

C# Cθ(u, v) ϕθ(t) parameter range

1 (Clayton) [u−θ + v−θ − 1]−1/θ ϕθ(t) [0, ∞) \ {0}

2 (Ali-Mikhail-Haq) uv1−θ(1−u)(1−v)

ln1−θ(1−t)

t[−1, 1]

3 (Gumbel-Hougaard) exp(−[(− ln u)θ + (− ln v)θ ]1/θ) (− ln t)θ [1, ∞)

4 (Frank) − 1θ

ln(1 +(e−uθ−1)(e−vθ−1)

e−θ−1− ln e−θt−1

e−θ−1(−∞, ∞) \ {0}

5 (Joe) 1 − [(1 − u)θ + (1 − v)θ − (1 − u)θ(1 − v)θ ]1/θ − ln[1 − (1 − t)θ ] [1, ∞)

6 (Gumbel-Barnett) uv exp(−θ ln u ln v) ln(1 − θ ln t) (0, 1]

7 uv

[1+(1−uθ)(1−vθ)]1/θln(2 − t−θ − 1) (0, 1]

8 (1 + [(u−1 − 1)θ + (v−1 − 1)θ ]1/θ)−1(

1t

− 1)θ

[1, ∞)

9 exp(1 − [(1 − ln u)θ + (1 − ln v)θ − 1]1/θ) (1 − ln t)θ − 1 (0, ∞)

10 (1 + [(u−1/θ − 1)θ + (v−1/θ − 1)θ ]1/θ)−θ (t−1/θ − 1)θ [1, ∞)

11 12

(S +

√

S2 + 4θ), S = u + v − 1 − θ( 1u

+ 1v

− 1)(

θt

+ 1)

(1 − t) [0, ∞)

12 (1 +[(1+u)−θ−1][(1+v)−θ−1]

2−θ−1)−1/θ − 1 − ln

(1+t)−θ−1

2−θ−1(−∞, ∞) \ {0}

13 θ/ ln(eθ/u + eθ/v − eθ) eθ/t − eθ (0, ∞)

14 [ln(exp(u−θ) + exp(v−θ) − e)]−1/θ exp(t−θ) − e (0, ∞)

Table 4: Overview of the (strict) Archimedean families (with generator) included in the bivariate

test space.

A.2 Other copula families

Next to the Archimedean copulas, we consider some extreme value copulas, some meta-elliptical

copulas and a few other copula models.


C# Cθ(u, v) parameter range

15 (Galambos) uv exp ((− log u)−θ + (− log v)−θ)− 1

θ θ ∈ [0, ∞)

16 (Husler and Reiss) exp(log(u)Φ( 1θ

+ 12

θ loglog ulog v

) + log(v)Φ( 1θ

+ 12

θ loglog vlog u

)) θ ∈ [0, ∞)

17 (Normal) Φρ(Φ(u)−1, Φ(v)−1) ρ ∈ [−1, 1]

18 (Student’s t) tρ,v(t−1v (u), t−1

v (v)) ρ ∈ [−1, 1]

19 (Farlie-Gumbel uv + uvθ(1 − u)(1 − v) θ ∈ [−1, 1]

-Morgenstern)

20 (Plackett)(1+(θ−1)(u+v))−

√

(1+(θ−1)(u+v))2−4uvθ(θ−1)

2(θ−1)θ ∈ (0, ∞)

21 (Raftery) min(u, v) + 1−θ1+θ

(uv)1/(1−θ)(1 − [max(u, v)]−(1+θ)/(1−θ)) θ ∈ [0, 1]

Table 5: Overview of the 7 other copula families included in the bivariate test space.

B. Multivariate copula families

B.1 Archimedean copula families

Exchangeable copulas, extensions from bivariate copula families.

Recall that a strict n-dimensional exchangeable Archimedean copula is defined as C : [0, 1]n →

[0, 1] : (u1, · · · , un) 7→ C(u1, · · · , un) = ϕ−1(∑n

i=1 ϕ(ui)), where ϕ : [0, 1] → [0,∞) is continuous,

strictly decreasing and convex, with ϕ(0) = +∞ and ϕ(1) = 0.

The inverse ϕ−1 has to be completely monotone, i.e. (−1)k dkϕ−1(t)dtk ≤ 0 for all k ∈ N0. Copulas

with this property are C1, C2, C3, C4, C5, C8, C9 and C10.

Partially exchangeable copulas, introduced by Joe (1997).

A simple multivariate generalization of the Archimedean copula in dimension d results in a depen-

dence structure of partial exchangeability. These copulas are referred to as fully nested, since a

higher dimensional copula is obtained by adding one dimension step by step.

For three dimensions, this results in

C(u1, u2, u3) = ϕ−12

[

ϕ2

(

ϕ−11

[

ϕ1(u1) + ϕ1(u2)]

)

+ ϕ2(u3)

]

.

In general, this copula can be written as

C(u1, · · · , ud)

= ϕ−1d−1

[

ϕd−1

(

ϕ−1d−2

[

ϕd−2

(

· · ·ϕ2(ϕ−11 [ϕ1(u1) + ϕ1(u2)]) + ϕ2(u3)])+

· · · + ϕd−2(ud−1)]

)

+ ϕd−1(ud)

]

.

(9)

The copula defined by (9) will be a proper d-copula if, in addition to the property of complete

monotonicity for the inverse functions of the generators, also the composite generators ϕi+1 ◦ ϕ−1i

are completely monotone.


For the three-dimensional case, Joe (1997) section 5.3 proposes to use four models of the form

Cθ1(u3, Cθ2

(u1, u2)) with θ2 ≥ θ1.

C# Cθ1(u3, Cθ2

(u1, u2)) θ2 ≥ θ1 τ12, τ13, τ23

range range

M3 −θ−11 log{1 − (1 − e−θ1 )−1(1 − [1 − (1 − e−θ2 )−1(1 − e−θ2u1 ) [0,∞) [0, 1]

·(1 − e−θ2u2 )]θ1/θ2 )(1 − e−θ1u3 )}

M4 [(u−θ21 + u−θ2

2 − 1)θ1/θ2 + u−θ13 − 1]−1/θ1 [0,∞) [0, 1]

M5 1 − {[(1 − u1)θ2 (1 − (1 − u2)θ2 ) + (1 − u2)θ2 ]θ1/θ2 · [1,∞) [0, 1]

·(1 − (1 − u3)θ1 ) + (1 − u3)θ1}1/θ1

M6 exp {−([(− log u1)θ2 + (− log u2)θ2 ]θ1/θ2 + (− log u3)θ1 )1/θ1} [1,∞) [0, 1]

Table 6: Overview of the 4 partially exchangeable copula families included in the trivariate test

space.

Note that model CM3 is an extension of C4, CM4 is an extension of C1, CM5 is an extension of C5

and CM6 is an extension of C3.

B.2 Multivariate Extreme value families

We use three multivariate extreme value families CMM1, · · · , CMM3 introduced in Joe (1997),

section 5.5. Note that the extreme value copulas are defined for positive dependence, implying

τ ∈ [0, 1].

For the trivariate case we have

• CMM1(u1, u2, u3) = exp

{

−[

((p1zθ1)δ12 + (p2z

θ2)δ12)1/δ12 + ((p1z

θ1)δ13 + (p3z

θ3)δ13)1/δ13

+((p2zθ2)δ23 + (p3z

θ3)δ23)1/δ23 + ν1p1z

θ1 + ν2p2z

θ2 + ν3p3z

θ3

]1/θ}

,

where θ ≥ 1, δij ≥ 1, νj ≥ 0, pj = (νj + 2)−1 and zj = − log uj .

• CMM2(u1, u2, u3) =

[(

u−θ1 + u−θ

2 + u−θ3

)

− 2 −

(

(u−δ121 + u−δ12

2 )−1/δ12

+(u−δ131 + u−δ13

3 )−1/δ13 + (u−δ232 + u−δ23

3 )−1/δ23

)]

−1/θ

,

where θ > 0, δij > 0, νj ≥ 0, pj = (νj + 2)−1 and uj = pj(u−θj − 1).

Note that the special case [u−θ1 + u−θ

2 + u−θ3 − 2]−1/θ (C1) arises for p1, p2, p3 reaching 0.


• CMM3(u1, u2, u3) = exp

{

−[(

zθ1 + zθ

2 + zθ3

)

−(

(p−δ121 z−θδ12

1 + p−δ122 z−θδ12

2 )−1/δ12

+(p−δ131 z−θδ13

1 + p−δ133 z−θδ13

3 )−1/δ13 + (p−δ232 z−θδ23

2 + p−δ233 z−θδ23

3 )−1/δ23

)]1/θ}

,

where θ > 1, δij > 0, νj ≥ 0, pj = (νj + 2)−1 and zj = − log uj .

Note that family CMM1 can be seen as an extension of C3, while CMM2 and CMM3 are extensions

of C15.

B.3 Meta-elliptical families

The meta-elliptical families can straightforwardly be extended to any dimension n ≥ 2.

B.4 MFGM family

We use the extension described in Nelsen (2006), page 108. In the trivariate case the extended

copula can be written as

CMFGM (u1, u2, u3) = u1u2u3[1 + θ12(1 − u1)(1 − u2) + θ13(1 − u1)(1 − u3)

+ θ23(1 − u2)(1 − u3) + θ123(1 − u1)(1 − u2)(1 − u3)]

where θ12, θ13, θ23, θ123 ∈ [−1, 1], which implies τ12, τ13 and τ23 ∈ [−2/9, 2/9].

C. Empirical copula

The empirical copula in d dimensions is a rank-based estimator calculated as

Cemp(u1, . . . , ud) =1

n

n∑

i=1

1(Ui1 ≤ u1, . . . , Uid ≤ ud) .

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Anew

gra

phica

lto

olfo

rco

pula

selection

25

Figure 8: 2D mapping of bivariate comparable test spaces for τ < 0.

−0.05 0 0.05−0.05

0

0.05

C4

C11 C

20

C17

C18

(a) τ = −90%

−0.1 −0.05 0 0.05 0.1−0.1

−0.05

0

0.05

0.1

C4

C11

C20

C17

C18

(b) τ = −80%

−0.15 −0.075 0 0.075 0.15−0.15

−0.075

0

0.075

0.15

C4

C14

C20

C17

C18

(c) τ = −70%

−0.6 −0.3 0 0.3 0.6

−0.6

−0.3

0

0.2

0.6

C4

C11

C12

C20

C17C

18

(d) τ = −60%

−0.5 −0.25 0 0.25 0.5−0.5

−.25

0

0.25

0.5

C4

C11

C12

C20

C17C

18

(e) τ = −50%

−0.5−0.2500.250.5−0.5

−0.25

0

0.25

0.5

C4

C11 C

12

C20

C17

C18

(f) τ = −40%

−0.5−0.2500.250.5−0.5

−0.25

0

0.25

0.5

C4 C

6

C9C

11 C12

C20

C17

C18

(g) τ = −30%

−0.5 −0.25 0 0.25 0.5−0.5

−0.25

0

0.25

0.5

C4

C6

C9 C

11C12 C

19

C20

C17

C18

(h) τ = −20%

−0.6 −0.3 0 0.3 0.6−0.6

−0.3

0

0.3

0.6

C2

C4

C6

C7C

9 C11

C12 C

19

C20

C25

C18

(i) τ = −10%

Anew

gra

phica

lto

olfo

rco

pula

selection

26

Figure 9: 2D mapping of bivariate comparable test spaces for τ > 0.

−0.5 −0.25 0 0.25 0.5

−0.5

−0.25

0

0.25

0.5

C1

C1S

C2

C3

C3S

C4

C5

C9

C11

C12

C14

C21

C19 C

20

C17

C18

C15

C16

(a) τ = 10%

−0.5 −0.25 0 0.25 0.5−0.5

−0.25

0

.25

.5

C1 C

1S

C2

C3

C3S

C4

C5

C9C

11

C12

C14C

21

C19

C20C

17

C18

C15

C16

(b) τ = 20%

−0.5 −0.25 0 0.25 0.5−0.5

−0.25

0

0.25

0.5

C1

C1S

C2

C3

C3S

C4

C5

C9

C11

C12

C14

C21

C20C17

C18

C15

C16

(c) τ = 30%

−0.5 −0.25 0 0.25 0.5−0.5

−0.25

0

0.25

0.5

C1

C1SC

3C

3S

C4

C5

C8

C9

C10

C12

C13

C14

C21

C20

C17

C18

C15

C16

(d) τ = 40%

−0.5 −0.25 0 0.25 0.5−0.5

−0.25

0

0.25

0.5

C1 C

1SC

3C

3S

C4

C5

C8

C9

C10

C12

C13

C14

C21 C

20

C17

C18

C15

C16

(e) τ = 50%

−0.5 −0.25 0 0.25 0.5−0.5

−0.25

0

0.25

0.5

C1

C1S

C3

C3S

C4

C5

C8

C9

C10

C12

C13C

14 C21

C20

C17

C18 C

15

C16

(f) τ = 60%

−0.5 −0.25 0 0.25 0.5−0.5

−0.25

0

0.25

0.5

C1 C

1SC

3

C3S

C4

C5

C8

C9

C10

C12

C13C

14 C21

C20

C17

C18

C15

C16

(g) τ = 70%

−0.5 −0.25 0 0.25 0.5−0.5

−0.25

0

0.25

0.5

C1

C1S

C3

C3S

C4

C5

C8

C9

C10

C12

C13C

14 C21

C20

C17

C18

C15

C16

(h) τ = 80%

−0.15 −0.075 0 0.075 0.15−0.15

−0.075

0

0.075

0.15

C1

C1S

C3

C3S

C4

C5

C8

C9 C

10C

12C13

C14

C21

C20

C17C

18

C15

C16

(i) τ = 90%

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